Protonium Formation in Antiproton–Hydrogen Collisions

We present a computation of the cross section of protonium formation in antiproton-hydrogen collisions using the nonlocal resonance model. It is based on restoration of the Born-Oppenheimer approximation, when the adiabatic electronic basis is replaced with the so called diabatic one. This allows us to separate the electron and hadron parts of our problem at the cost of a nonlocal potential in hadron dynamics. Introduction Many groups used different methods to describe collisions of a hydrogen atom and negatively charged heavy particles. The first article about the formation of an atom which has a heavy particle instead of an electron was written by Fermi and Teller. In their work (studying the capture of heavy negative particles in matter) [Fermi and Teller, 1947] they also deduced the critical distance between the proton and the heavy particle now called the Fermi-Teller radius. If the distance is smaller then this radius then the potential is not strong enough to hold the electron bound. This means that we can not use the adiabatic approximation, because all electronic bound states disappear in continuum (see figure 1). More sophisticated methods must be used. Older works are summarized in [Cohen, 2004]. Advanced adiabatic methods were used by Ovchinnikov and Macek [2005] and by Žďánská et al. [2004]. Hesse et al. [2004] employed the hyperspherical close coupling method. The wave packet evolution and semiclassical approach were used by Sakimoto [2002a, 2001, 2002b]. Yamanaka and Ichimura [2006] solved Faddeev equations for this three body system and calculated probabilities of protonium formation. Some of the approaches scale down the mass of the antiproton to be able to use their method and get some qualitative results, because the antiproton is captured in states with n ≈ 40 and l remains approximately constant in the collision. The reaction p̄ + H → pp̄ + e is also a prototype of the associative detachment reaction studied in atomic physics [Bieniek and Dalgrano, 1979; Čı́̌zek et al., 1998]. As a one electron system, free of electron correlation, it is vital for the understanding of errors that are caused by some approximations used in such calculations. Next, we introduce the theoretical background of our calculation of hadron dynamics during the collision and show some results for a preliminary fit to complex level–shift which we took from our previous calculation for electron motion with fixed hadrons. Theory The nonlocal resonance model for collisions of negative particles with an atom was developed elsewhere (see [Domcke, 1991; Čı́̌zek, 1999]) and here we summarize it. The Hamiltonian of our problem is (we use atomic units unless stated otherwise) H = Tpp̄ +Hel, Tpp̄ = P 2 2μ , Hel = p 2 2 − 1 rp−e + 1 rp̄−e − 1 R , (1) where p and P denote the momenta of the electron and hadrons in the center of mass system respectively. We introduce an expansion of the wave function